p.1065

A Generalization of the Quasilinear Mean with Applications to the Measurement of Income Inequality and Decision Theory Resolving the Allais Paradox

Chew Soo Hong

Abstract

The main result of this paper is a generalization of the quasilinear mean of Nagumo [29], Kolmogorov [26], and de Finetti [17]. We prove that the most general class of mean values, denoted by M"@?@?, satisfying Consistency with Certainty, Betweenness, Substitution-independence, Continuity, and Extension, is characterized by a continuous, nonvanishing weight function a? and a continuous, strictly monotone value-like function @?. The quasilinear mean M"@? results whenever the weight function @? is constant. Existence conditions and consistency conditions with first and higher degree stochastic dominance are derived and an extension of a well known inequality among quasilinear means, which is related to Pratt's [31] condition for comparative risk aversion, is obtained. Under the interpretation of mean value as a certainty equivalent for a lottery, the M"@?@? mean gives rise to a generalization of the expected utility hypothesis which has testable implications, one of which is the resolution of the Allias "paradox." The M"@?@? mean can also be used to model the equally-distributed-equivalent or representative income corresponding to an income distribution. This generates a family of relative and absolute inequality measures and a related family of weighted utilitarian social welfare functions.